Any way to send a picture via PM in the chat?

Seem to be not possible so far. Irks me a bit too.

You could always upload it somewhere else and send a link...

But make sure that the recipient is informed beforehand that you are gonna send them a link with the picture and what it is. Nobody should be clicking links in PMs they don't know where they're leading to, even if they seem innocent. ;)

Why would you even need to? It just clutter the chat history and it wastes space/bandwidth (among other things).

As already mentioned you can just use a free/public image service. If that's not an option, well, let's just say GOG isn't THAT kind of sharing place ;)

I guess it's also to protect themselves for unwanted attention from *insert local enforcement or european laws* in case someone decided to share something that isn't game related. They don't seem to have the capacity to handle the forum sufficiently, or the amount support tickets, why would they give themselves more trouble to handle it or to clean up the mess? Adding manual or auto scanning software/algorithms costs lots of money and resources.

EDIT: Also, by proxy, GOG used to stand for good old games; there was a time were you only had text based games, so I guess it's kinda fitting to have a minimalist text only chat/forum (with the exception of the user icon of course) :)